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Optimizing Daily Operation of Battery Energy
Storage Systems under Real-Time Pricing Schemes
Juan M. Lujano-Rojas, Rodolfo Dufo-López, José L. Bernal-Agustín, Member, IEEE,
and João P. S. Catalão, Senior Member, IEEE
μ

Abstract--Modernization of electricity networks is currently
being carried out using the concept of the Smart Grid; hence, the
active participation of end-user consumers and distributed
generators will be allowed in order to increase system efficiency
and renewable power accommodation. In this context, this paper
proposes a comprehensive methodology to optimally control leadacid batteries operating under dynamic pricing schemes in both
independent and aggregated ways, taking into account the effects
of the charge controller operation, the variable efficiency of the
power converter, and the maximum capacity of the electricity
network. A genetic algorithm is used to solve the optimization
problem in which the daily net cost is minimized. The
effectiveness and computational efficiency of the proposed
methodology is illustrated using real data from the Spanish
electricity market during 2014 and 2015 in order to evaluate the
effects of forecasting error of energy prices, observing an
important reduction in the estimated benefit as a result of both
factors: forecasting error and power system limitations.
,
α
(α)
̅
Index Terms—Smart Grid, lead-acid battery, electricity price
forecasting, battery energy storage system, real-time pricing.
I. NOMENCLATURE
Index for AR parameters ( = 1, … , ).
Index for MA parameters ( = 1, … , ).
Index for each element of HWEP ( = 1, … , ).
Index for individuals of GA ( = 1,2, … , ).
Index for day time ( = 1, … , ).
Index for BESS ( = 1, … , ).
HWEP time series (€/kWh).
Transformed HWEP.
Transformed and standardized HWEP.
FEP daily series (€/kWh).
Transformed FEP.
Transformed and standardized FEP.
This work was supported by FEDER funds through COMPETE and by
Portuguese funds through FCT, under FCOMP-01-0124-FEDER-020282
(PTDC/EEA-EEL/118519/2010),
PEst-OE/EEI/LA0021/2013
and
SFRH/BPD/103079/2014. Moreover, the research leading to these results has
received funding from the EU Seventh Framework Programme FP7/2007-2013
under grant agreement no. 309048. This work was also supported by the Ministerio
de Economía y Competitividad of the Spanish Government under Project
ENE2013-48517-C2-1-R.
Juan M. Lujano-Rojas is with the University of Beira Interior, Covilhã 6201001, Portugal, and also with INESC-ID, Instituto Superior Técnico, University of
Lisbon, Lisbon 1049-001, Portugal (e-mail: [email protected]).
Rodolfo Dufo-López and José L. Bernal-Agustín are with the University of
Zaragoza, Zaragoza 50018, Spain (e-mail: [email protected]; [email protected]).
João P. S. Catalão with INESC TEC and the Faculty of Engineering of the
University of Porto, Porto 4200-465, Portugal, and also with C-MAST, University
of Beira Interior, Covilhã 6201-001, Portugal, and also with INESC-ID, Instituto
Superior Técnico, University of Lisbon, Lisbon 1049-001, Portugal (e-mail:
[email protected]).
̅
,
/
,
Hourly mean of transformed HWEP (€/kWh).
CDF of a normalized Gaussian PDF.
CDF of HWEP.
Inverse CDF of a normalized Gaussian PDF.
Inverse CDF of HWEP.
DNC of individual for BESS at time (€).
Autoregressive parameters of ARMA model.
Moving average parameters of ARMA model.
Box-Pierce statistics
Significance level of statistical test
α quantile of chi-squared distribution.
Number of lags of statistical analysis.
Minimum SOC of BESS system
SOC of BESS at time .
DOD of BESS at time .
Battery capacity of BESS in 10h (Ah/cell).
Normalized capacity in respect to a 100 Ah
cell.
Battery voltage of BESS at time (V/cell).
Open circuit voltage at full SOC (2.1 V/cell).
Variation of voltage with SOC (0.076 V/cell).
Normalized gassing voltage (2.23 V/cell).
Regulation battery voltage (2.23 V/cell).
Estimation error of BM (V/cell).
Error of BM at left side of interval (V/cell).
Error of BM at right side of interval (V/cell).
Voltage at left side of interval (V/cell).
Voltage at right side of interval (V/cell).
Battery current of BESS at time (A/cell).
Battery current of BESS in 10h (A/cell).
Maximum battery current of BESS (A/cell).
Normalized gassing current (20 mA/cell).
Gassing current of BESS at time (A/cell).
Current at left side of search interval (A/cell).
Current at right side of search interval (A/cell).
Intermediate BM variables (A/cell).
Value of a time step (1h).
Normalized gassing temperature (298 K).
Ambient temperature of BESS at time (K).
Normalized battery capacity (1.001).
Normalized battery capacity (1.75).
Voltage factor of gassing process (11V-1).
Temperature factor of gassing (0.06V-1).
Parameter of converter (0.078815).
Parameter charge-transfer process (0.888).
Parameter charge-transfer process (0.0464).
Parameter of converter (0.015784).
Charging internal resistance (0.42 Ω Ah).
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,
,
,
,
,
,
⃗
Discharging internal resistance (0.699 Ω Ah).
Number of cells in serial of BESS .
Number of cells in parallel of BESS .
Power from/to LAB at time (kW).
Power from/to BESS at time (kW).
Rated power of converter of BESS (kW).
Converter efficiency of BESS .
Power of BESS aggregator at time (kW).
Reduced power setting per cell (W/cell).
Power of individual for BESS at time
(kW).
Optimal scheduling of BESS at time (kW).
Population of GA.
Population size of GA.
Number of generation of GA.
Crossover rate of GA (%).
Mutation rate of GA (%).
Individual of GA.
Control action for individual at time .
Fitness of individual for BESS .
Maximum capacity of SG (kW).
Overload of SG (kW).
Weighting factor of BESS .
Maximum power per cell of BESS (W/cell).
Reduced power per cell of BESS (W/cell).
Iterations of BM ( = 1, … , ).
II. INTRODUCTION
S a result of constant industrialization, technological
development, and the economic growth or many
countries, social awareness of environmental problems has
increased the incorporation of renewable energies for power
generation. However, technical limitations in transmission and
distribution networks, the variability of renewable power
sources such as wind and solar photovoltaic (PV) energies,
people’s behaviors, and energy consumption patterns have
made it difficult to integrate and accommodate these sources.
In this scenario, the integration of storage technologies has
emerged as an option to expedite energy consumption from
renewable sources by increasing the flexibility of the power
system. Topics related to the Battery Energy Storage System
(BESS) operation need to be considered to guarantee the
feasible integration of a BESS with renewable and
conventional generation sources currently being operated; this
includes the estimation of storage capacity and its
corresponding duration—or, in other words, the estimation of
the available power and discharge time—and the incorporation
of the transmission and distribution capacity [1]. With the
conceptual evolution of traditional distribution systems toward
the Smart Grid (SG) and, consequently, the implementation of
dynamic pricing schemes such as Time of Use (TOU) and
Real Time (RT) programs, incorporating a BESS at the
residential level would improve power system performance;
however, some specific techno-economic conditions are still
required. According to recent studies, under TOU [2] and RT
[3] programs, the acquisition costs of a BESS have to be
reduced, its lifetime in terms of the number of cycles should
be increased, and the difference between the maximum and
A
minimum daily electricity prices should be increased in order
to compensate capital and operational costs. In this regard,
governmental policies and incentives are very important. A
representative example is the Self-Generation Incentive
Program [4] administrated by the California Public Utilities
Commission. This program aims to incentivize electricity
production from wind turbines, fuel cells, combined heat and
power generation systems, and advanced energy storage
systems (ESSs).
The optimal operation of ESSs, considering an RT pricing
tariff, becomes particularly relevant for the successful
adoption and integration of these technologies; this problem
has been analyzed from different perspectives by several
authors, and the results obtained from their efforts can be
found in the specialized literature.
In [5], historical data on the electricity price forecasting of
the Ontario market are carefully analyzed in terms of
forecasting error and its impact on the profitability of a
Compressed Air Energy Storage (CAES) system operating
under dynamic pricing. This analysis is carried out by solving
a Mixed-Integer Linear Programming (MILP) problem, in
which the economic revenue obtained from the operation of
the CAES system is maximized and consideration is given to
its corresponding operational constraints. Then, in order to
mitigate the negative effects of forecasting error, the objective
function is calibrated by taking into account the magnitude of
the prediction error that occurred during the last hours or days.
In [6], a scheduling procedure to improve renewable power
accommodation by simultaneously reducing power
curtailment and forecasting error was presented and analyzed
through mathematical simulations. The method consists of two
main steps, a scheduling process on a daily basis and real-time
control. During the scheduling process, an optimal day-ahead
profile of the joint system (renewable generator and ESS) is
determined by considering the energy actually stored on the
ESS, renewable power, and load demand predictions. During
the real-time control process, the power of the joint system is
dispatched in order to follow the optimal generation profile
determined in the scheduling procedure, taking into account
all the operational constraints of the joint system. According
to the obtained results, this control strategy offers an important
reduction in forecasting error even when ESSs of moderate
capacity are considered.
In [7], a control strategy for the operation of an energy
system provided with PV generation and a BESS installed at
the residential level was presented considering three price
periods (off-peak period, low-peak period, and high-peak
period). This approach uses predictions of PV generation and
load demand, which are incorporated into a hierarchical
control system composed of two main levels, a global one and
a local one. The global level optimally determines the
condition (charging or discharging) of BESSs in the future
based on cost minimization and peak shaving, while the local
level reduces the impact of renewable power and load demand
forecasting errors. In a similar way, a control algorithm that
analyzes BESS management from an intra-daily perspective
was developed in [8]. The control strategy aims to minimize
the total electricity cost by solving the corresponding optimal
storage control problem using a piecewise linear
approximation of the objective function embedded in a Linear
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Programming (LP) problem. In order to guarantee the minimal
storage requirements of the BESS under analysis, a
reinforcement learning technique is incorporated.
In [9], the concept of an aggregator for PV-BESSs installed
at the residential level, including physical and communication
infrastructures, was developed. The aggregator is assumed to
have a centralized operation mode, where its optimal behavior
is determined through the solution of a MILP problem carried
out in two steps. In the first step, integer variables related to
Electric Vehicle (EV) charging are found by using a
continuous relaxation method; in the second step, the optimal
values for the rest of the variables are determined by solving
an LP problem. Besides this, the forecasting error of electricity
prices, PV generation, and EV charging profile are handled by
using Model Predictive Control (MPC).
In [10], cooperative-driven, distributed MPC was proposed
as a control method for BESSs in order to perform voltage
regulation in distribution networks. This method is based on
the solution of an optimization problem in which the
variations on the control variables are minimized. According
to the reported results from the analysis of a typical rural
distribution network provided with renewable generation
under over-voltage, low-voltage, and voltage-drop conditions,
the presented approach offers similar results to those obtained
from the implementation of the centralized MPC method.
The researchers in [11] deeply analyzed the behavior of a
typical energy system composed of a conventional generator,
several base and flexible loads, several subsystems composed
of renewable generators and ESSs, and several external
electricity markets. Optimal operation is mathematically
formulated in order to minimize the system cost on a longterm basis through the manipulation of controllable energy
sources, load demand, and ESSs, taking into account the
uncertainty introduced by renewable power sources. This
optimization problem is solved using the Lyapunov
optimization technique combined with a relaxation procedure.
Considering the energy policy in which residential
consumers are only able to buy electricity from the SG, a
BESS control strategy based on a Dynamic Programming
(DP) approach was developed in [12]. The presented
methodology considers specific discrete states for charging
and discharging conditions; in order to avoid those solutions
that could negatively influence a BESS’s lifetime, a backward
tracing procedure is introduced. Finally, corrective control
actions are taken during real-time operation so that the effects
of forecasting error of electricity prices, load demand, and
renewable generation are mitigated.
In [13], the implementation of a DP approach in two
temporal scales was proposed for the control of energy
systems with a BESS, typical residential loads, and renewable
power generation under dynamic pricing. The presented
method consists of the application of a DP optimization
method on the macro- and microscales. A macroscale DP
method is applied by considering the forecasting information
on a daily basis; hence, a first approximation of the optimal
scheduling is obtained in terms of the State of Charge (SOC)
of the BESS. Then, the microscale DP method is applied using
forecasting information with a shorter prediction horizon
(three-hour-ahead predictions) in order to guarantee the
optimal behavior expressed in terms of the SOC obtained in
the previous step; in this way, the impact of the forecasting
error can be mitigated.
In [14], a strategy based on a DP approach for the control of
a Lead-Acid Battery (LAB) was developed; the DP approach
was applied by considering discrete states of the SOC with the
objective of maximizing the economic benefits from the daily
operation of the BESS, taking into account the Battery Wear
Cost (BWC), full filling of certain operational constraints
related to SOC limits, and charging and discharging rates.
Lifetime reduction of the BESS related to operation at partial
SOC is incorporated as an increment in system costs; this is
known as the Cost of Lifetime Losses estimation.
Assuming that residential consumers are only able to buy
electricity from the SG, the rated power of the BESS is
applied during charging and discharging periods of equal
duration, as well as the representation of the SOC by a number
of limited discrete states; in [15], a control strategy was
developed for the optimal operation of a BESS on semi-daily
and daily bases, defined according to the Moving Average of
RT pricing forecasting.
In [16], the operation of BESSs installed over distribution
networks was analyzed and their capabilities to mitigate
voltage fluctuations were studied through numerical
simulations. Assuming that the Distribution Network Operator
(DNO) is able to give subsidies to commercial customers to
promote the local installation of ESSs, the storage capacity at
the customer level is determined by taking into account the
aforementioned subsidies and high-voltage distribution
network constraints, specifically those related to voltage
variations. This approach allows us to consider the viewpoint
of the DNO and commercial customers at the same time.
In [17], a methodology based on the implementation of
hyper-heuristic algorithms was proposed. Considering the
maximum possible power to be supplied, 26 heuristic levels—
from no discharging condition to totally discharging
condition—were defined; then, the corresponding fitness
function and heuristic operators were applied in order to
determine a profitable discharging scheduling for the ESS
under analysis. The proposed methodology was illustrated by
considering a Vented LAB, whose dynamic behavior
regarding the capacity as a function of discharge time and
charging power limitations were incorporated by using
general-purpose information normalized with respect to the
capacity in 240h.
In [18], an optimization algorithm for the optimal operation
of ESSs based on power-to-heat conversion was presented.
The fundamental idea consists of charging ESSs during lowprice periods, taking into account the amount of energy stored
and the maximum charging power allowed at each time step.
Initially, a mathematical formulation neglecting energy losses
is developed; then, two optimization models considering
energy losses as a constant and variable value are derived.
In [19], a management model for BESSs integrated in a
smart community was analyzed by developing an auction
process by which each owner of a BESS, customers, and
building managers are related to each other through a market
environment. The storage capacity and its corresponding
auction price are optimally determined by means of a
noncooperative Stackelberg game established between the
owners of BESSs and building managers acting as auctioneers.
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In this way, the incentive compatibility and individual
rationality are compensated at the equilibrium point of the
Stackelberg game.
A control strategy based on the theory of pseudo-SOC was
developed in [20] to incorporate ancillary services provided by
distributed generators; the algorithm takes information from
the operation of the energy system, such as an SOC, as well as
the amount of PV generation, in order to estimate the real-time
operating conditions in terms of voltage and determine the
optimal operating strategy of the BESS.
In [21], several resources, strategies, and operational
problems such as renewable power generation, peak-load
shaving, power-curve smoothing, and voltage regulation have
been integrated in a control technique based on minimizing
total costs, by which the costs related to the peak-load shaving
activity are weighted according to the amount of power
flowing through the distribution transformer. Using this
formulation, the corresponding optimization problem is solved
by means of a sequential quadratic programming approach.
Among ESS technologies, advanced LABs (specifically
Valve-Regulated LABs) have been installed and analyzed in
terms of their technical design and performance in order to
gain experience and knowledge of their operational problems,
such as thermal management issues and cell behavior [22].
According to the Global Energy Storage Database [23],
BESSs based on LABs are currently being investigated to
improve the accommodation of renewable generation, manage
and reduce electricity bills, shift energy time, enhance power
supply and spinning reserve, regulate frequency, improve
system reliability and the power quality of commercial
consumers, and enhance power-system ramping capabilities,
as well as renewables’ capacity firming. The typical rated
power of these installations is between 2 kW and 36 MW.
They are currently being operated in several countries,
including Australia, Cape Verde, Germany, India, Ireland,
Japan, Kenya, South Korea, Madagascar, New Zealand, South
Africa, Taiwan, the United Kingdom, and the United States,
among others, and administrated by cooperatives and
independent investors, as well as by federal and municipal
institutions.
Although many of the existing methodologies for the
optimal control of a LAB, when it is operating under dynamic
pricing, offer reasonable solutions from a mathematical
perspective, many of them are not capable of incorporating the
nonlinear behavior of the LAB, such as the gassing
phenomena of electrochemical processes, the voltage
regulation of the charge controller, and the variable efficiency
of the bidirectional converter. On the one hand, during the
charging process of a typical LAB, the charge controller
reduces the corresponding current in order to mitigate the
effects of the gassing phenomenon by maintaining the battery
voltage at a constant value (frequently 2.23 V/cell); as a
consequence, the amount of power purchased from the grid is
gradually reduced in a nonlinear way. On the other hand, the
efficiency of the power converter has nonlinear behavior:
When a low amount of power (compared to its rated capacity)
is transmitted through the converter, the conversion efficiency
is very low; however, this efficiency increases with the
transmitted power. This fact could influence the power trading
between a LAB and the SG and, consequently, the economic
performance of the system.
In this paper the optimal operation of LABs, including all
these issues, is analyzed from the perspective of an
independent BESS and an aggregator of it. As an optimization
tool, an integer-coded Genetic Algorithm (GA) is used,
considering charging, discharging, and no-power transactions
between the BESS and the SG. This tool is used to analyze the
influence of the electricity price forecasting error and the
limitations of the SG (in terms of the maximum power
allowed to be taken from or supplied to the SG by the BESS
aggregator) on the economic benefit obtained from the optimal
BESS operation. The rest of the paper is organized as follows:
Section III describes the proposed approach, including the
electricity price forecasting process, LAB model, and
optimization technique. Section IV presents the case study
analyzed in order to illustrate the proposed approach. Finally,
the main conclusions are presented and discussed in section V.
III. PROPOSED APPROACH
In Fig. 1, the general structure of an independent BESS
connected to the SG under an RT pricing environment is
described. The SG is represented by the power system with
two main communication channels: one to the power flow
(Power in Fig. 1) in a bidirectional way and another one to the
data flow (Data in Fig. 1) specifically for the transmission of
electricity prices in real time. A Smart Meter (SM) has the role
of interfacing between the Battery Management System
(BMS) and the SG. The SM receives electricity prices and
measures the power transaction from or to the SG and
communicates this information to the BMS; then, all the
collected data related to electricity prices are used to create a
database of Historical Wholesale Electricity Price (HWEP)
required to carry out the Forecasting Electricity Price (FEP)
on a daily basis. Besides storing the HWEP database, the BMS
is continuously monitoring the operating conditions of the
LAB in terms of battery voltage, current, and SOC
(Voltage/Current/State of Charge in Fig. 1), and it is able to
store the parameters of the LAB model, which is very
important information to be jointly used with the FEP to
determine the optimal LAB scheduling through an
optimization analysis (using an integer-coded GA). This
optimization analysis is supposed to be implemented in the
BMS; hence, this unit could be conceptually understood as an
embedded system, which receives the FEP and the operating
conditions of the LAB in order to estimate its optimal
management during the following day. The optimal LAB
schedule consists of when the BESS should be charged or
discharged, including the corresponding charging and
discharging power, in order to maximize the economic benefit
obtained from the power transaction between the SG and the
BESS; this schedule is implemented by sending a control
signal from the BMS to the Charge Controller
(Charging/Discharging in Fig. 1). In brief, in the process
required to determine and execute the optimal schedule for the
BESS, it is necessary to carry out three main procedures:
electricity price forecasting, optimization analysis and LAB
simulation, and scheduling implementation. In the following
subsections, these procedures are carefully described and
discussed.
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Voltage / Current / State of Charge
=
Charging / Discharging
+
ɛ
⦡ = 1, … , ;
(4)
Power
Battery
Management System
Power
System
SM
=
BMS
Smart
Meter
LAB
=
Day-ahead Power
Charge Lead-Acid
Forecasting Converter Controller Battery
Data
Optimization
Analysis
Fig. 1. General description of an independent BESS connected to SG.
A. Day-Ahead Electricity Price Forecasting
In order to determine the optimal scheduling of the BESS
that would maximize the economic benefit of the power
transaction between the BESS and the SG, an FEP is
absolutely required. In this paper, a time-series analysis based
on an Autoregressive Moving Average (ARMA) model has
been used.
The ARMA model is implemented in the BMS and requires
the estimation of autoregressive and moving-average
coefficients. To carry out this task, the HWEP time series
needs to be manipulated to have the characteristics of a
normalized Gaussian probability distribution function (PDF).
This transformation process [24] is briefly described in (1),
where each element of the HWEP ( ⦡ = 1, … , ) is
evaluated on its own discrete Cumulative Distribution
Function (CDF) ( ), resulting in a new time series with a
uniform PDF. This time series is consequently evaluated (in
an inverse way) on the CDF of a discrete normalized Gaussian
PDF (
), which is obtained from the discretization
methodology presented in [25].
This procedure results in the time series ( ⦡ = 1, … , ),
which has a diurnal non-stationarity that is directly related to
the pattern of electricity usage. This is removed by subtracting
the hourly mean (μ ⦡ = 1, … ,24), as shown in (2). Then,
the ARMA model of (3) [26] is fitted by means of the
Bayesian Information Criterion (BIC) [27] or the Akaike
Information Criterion (AIC) [28] and is statistically verified
through the Ljung-Box method [29], which compares the
value of a chi-square distribution ( ) with the significance
level and the − − degree of freedom with the value of
statistic ( ).
=
=
⦡ = 1, … , ;
− μ ⦡ = 1, … , ⦡ = 1, … , ;
+ μ ⦡ = 1, … , ;
(1)
⦡ = 1, … , .
+
ɛ
⦡ = 1, … , .
B. Lead-Acid Battery Model and Simulation
The general-purpose LAB model presented in [30] is used
in this work due to its ability to represent physic-chemical
processes. As the model developed in this work could be used
to optimize the operation of a LAB aggregator, the
mathematical model is expressed in terms of an independent
BESS ( ). Current-voltage characteristics are shown in (7) and
(8) for charging and discharging conditions, respectively. This
is a modified version of the Shepherd model, which takes into
account changes of open-circuit voltage with the SOC, ohmic
losses related to grid resistance, active mass resistance, the
resistance of the electrolyte, and the battery overvoltage.
=
−(
⦡
=
−(
(2)
(3)
Day-ahead forecasting is carried out by evaluating the
ARMA model previously estimated, as shown in (4), and by
completing the transformation process as presented in (5)
and (6).
(6)
In brief, the entire process for day-ahead price forecasting
starts by creating the discretized CDF of the HWEP and the
Gaussian CDF; then, each element of the HWEP time series is
evaluated on its own CDF, and it is evaluated again on the
inverse Gaussian CDF. When all elements have been
evaluated, the hourly average profile is subtracted; thus, a
normalized time series is obtained. The next step consists of
determining the order and coefficients of the ARMA model.
Once this step has been concluded, the estimated model is
sequentially evaluated from =1 until = , the hourly average
profile is added, and the obtained result is transformed from a
discretized Gaussian PDF to the PDF of the HWEP by
evaluating each of these values (those values resulting from
the evaluation of the ARMA model previously performed) on
the discretized Gaussian CDF and again on the inverse CDF of
the HWEP.
⦡
=
(5)
)
+
+
−
> 0 ⦡ = 1, … , ;
)
+
−
≤ 0 ⦡ = 1, … , .
(7)
+
(8)
The relationship between the gassing current and the SOC
during charging conditions is shown in (9) and (10),
developed from the Butler-Volmer characteristic. In (9), the
reader can note how the model considers only those current
values whose associated energy is effectively stored on the
LAB by subtracting the current related to the gassing process
( , ) from the external charging current ( ).
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−
+
,
ALGORITHM I
ESTIMATION OF BATTERY CHARGING C URRENT
⦡ = 1, … , ; (9)
Step 1: Set an estimation error of BM (
= ( ̅ )( ̅ )
[ − ]+
⦡ = 1, … , .
,
−
(10)
Otherwise, in (10), the reader can note how the gassing
current increases exponentially with the cell voltage and
temperature, which justify the exponential reduction of the
coulombic efficiency at high SOC values. During discharging
conditions, the SOC could be estimated by applying (9),
neglecting the effects of the gassing phenomena expressed in
(10) [31]. In addition, it is important to consider each SOC
value between its minimum value and 1 (
∈
[
, 1]; = 1, . . . , ; = 1, … , ); the charge controller
technically carries this out to protect the battery cell against
over-discharging conditions. To guarantee the safe operation
of the BESS, the charging ( > 0) and discharging ( < 0)
currents are limited to determined values suggested by the
battery manufacturer; this idea is presented in the constraint
(11),
−
≤
≤
⦡ = 1, … , .
(11)
During charging conditions, the battery current is defined in
the interval (0, ], so that the current value at which the
battery voltage is close to its corresponding regulation value
( ) (which is established to protect the battery cell against
over-charging conditions) is obtained from the application of
the bisection method (BM) [32], according to Algorithm I.
During discharging conditions, the battery current is defined
in the interval [− , 0) so that it can be estimated from the
energy stored in the battery, which is available to be
discharged in a single time step ( ); this is expressed in (12):
=−
(
)
−
,
⦡ = 1, … , .
(12)
Finally, those situations with no power transaction between
the LAB and SG are simulated by assigning =0. Figs. 2 and
3 illustrate the results obtained from the simulation of a
battery cell of 300 Ah ( =300 Ah) when the battery current
is limited to 10A ( =10 A) and 30A ( =30 A), respectively.
It is possible to observe how the charging current is
exponentially reduced to maintain the battery voltage at the
corresponding regulation value, which directly influences the
power taken from the grid. In addition, the reader can observe
how the maximum power to be taken from the grid can be
reduced by using only the variable , according to Fig. 3,
from 60W to 20W; this fact is used to limit the amount of
power to be taken by or injected into the SG to preserve its
safe operation. Once a battery cell has been analyzed, all
variables are scaled to consider the total capacity of the battery
bank as presented in (13):
,
=(
)
.
(13)
Step 2: Set the maximum number of iterations ( ) at a value
( /
)⁄ (2).
higher than
Step 4: Set
⟵ .
⟵ 0 and
Step 3: Set
⟵ 1.
Step 4.1: Set ⟵ ; calculate
from the evaluation
of
on (7), (9), and (10); and the corresponding error is
estimated as
=
−
.
)/2; calculate
Step 4.2: Set ⟵ ( +
from the
evaluation of on (7), (9), and (10); and the corresponding
error is calculated as
=
−
.
)(
Step 4.3: If {(
⟵ .
Step 5: If { <
}, then
) > 0}, then
⟵
⟵
, else
+ 1; go to Step 4.1 or stop.
Then, the variable efficiency of the bidirectional converter is
incorporated using (14) and (15).
,
=
(
,
) + (1 +
=±
−
(1 +
,
)
;
(14)
,
( )
.
)
(15)
Power from/to LAB ( , ) is estimated by (15) only when the
power from/to LAB ( , ) is higher than factor (
) and
when , is lower than the rated power of the bidirectional
converter ( ). In other cases, , is assumed to be zero.
Corresponding parameters for modeling efficiency variations
have been obtained using information taken from [33].
2.24
2.22
35
30
Voltage
signal
2.2
Voltage (V/cell)
,
).
25
2.18
2.16
20
2.14
15
2.12
2.1
10
Current
signal
2.08
2.06
0:00
5:45
11:30 17:15
Time (h)
Current (A/cell)
=
5
0
23:00
Fig. 2. Controlled voltage and current of a typical battery cell.
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70
Power (kW)
60
Power
50
40
Power
30
20
10
0
0:00
5:45
11:30
Time (h)
17:15
23:00
Fig. 3. Controlled power of a typical battery cell.
C. Optimization of Day-Ahead Operation
As stated before, an integer-coded GA has been used in this
paper to determine the optimal operation of a LAB enrolled in
an RT pricing program on a daily basis. In the implementation
used in this work, each individual is represented by an integer
···
···
vector with =24 elements, such as this: ⃗ =(
) with =1,…, .
Each element ( ) corresponds to the operative condition of
the LAB; if =1, the charging of the LAB is carried out;
otherwise, if =−1, the discharging of the LAB is done; and
finally, if =0, there is no power transaction between the
BESS and SG; it is the equivalent of disconnecting the BESS
from the SG; in this way, the population of the GA is
presented in (16) and (17):
⃗
⎡
⎢⋮
= ⎢⃗
⎢
⎢⋮
⎣⃗
⃗ =[
…
⎤
⎥
⎥;
⎥
⎥
⎦
…
(16)
number of generations ( ), the crossover rate ( ), and the
mutation rate ( ) have been defined, the algorithm could be
implemented by following the steps of Algorithm II.
In this work, the number of possible combinations of
individuals is 324 =2.82·10 11. Evaluating all of them would
take years of computing, which is obviously inadmissible.
Therefore, it is impossible to know if the best solution found
by the GA is really the true optimal solution. However, we can
be confident that the optimal solution, or a near-optimal
solution, will be found.
The three steps described in Algorithm I are sequentially
repeated until the maximum number of generations ( ) is
reached; after the analysis of all of the generations, the
individual with the highest fitness value is chosen as the
solution to define the near-optimal schedule of BESS ( , ).
The fitness function of (19) establishes a linear relationship
among all individuals due to the fact that it is defined by the
position of the individual in the sorted list instead of the value
of the objective function (DNC), which could have important
differences between them.
The simulation process and evaluation of a determined
individual ( ⃗ ) could be understood easily by analyzing Fig. 4.
As can be observed, at =1h, ⃗ =−1 and the LAB should be
discharged so that the battery voltage is estimated by using
(8), the SOC is estimated by using (9) with , → 0, and the
discharging current is estimated according to (12). Now, at
time , ⃗ =1 and the LAB should be charged; hence, the
battery voltage is estimated by using (7), the SOC is estimated
by using (9) and (10), and the charging current is estimated by
using Algorithm I. Finally, at = , ⃗ =0, and there is no power
transaction between the BESS and SG. Then, the battery
voltage is estimated through (8), and
=
. Once
the BESS power has been calculated from the simulation
process previously described, the DNC could be estimated
easily for the individual under analysis by means of (18).
1: Charging
] ⦡ = 1, … , .
-1: Discharging
(17)
0: No power transaction
In Fig. 4, it is shown how a determined individual ( ⃗ ) is
included in the population ( ), which is expressed as a matrix.
The optimization algorithm aims to minimize the Daily Net
Cost (DNC). For a determined individual ( ⃗ ) in the
population, DNC is estimated according to (18):
,
=
( )
,
,
⦡ = 1, … , .
(18)
-1 -1 -1
1
1
1
0
0
0
1
··· ···
t
··· ···
T
2
Individual ( )
Population of GA (B)
-1
1
-1
0
1
1
0
0
0
1
1
-1
1
1
1
1
0
1
0
2
1
-1 -1
1
1
1
0
0
1
3
···
···
···
···
,
-1 -1 -1
1
1
1
0
0
0
i
1
1
0
1
1
1
0
0
0
···
The variable
is related to vector ⃗ through the
,
simulation process of III-B. In other words, the values of , ,
are obtained from the simulation process of (7)-(12) and
Algorithm I, and they are equal to the power from/to LAB
( , , = , ⦡ = 1, … , ).
This means these factors are not represented in our dailybased optimization model. Once the population size ( ), the
3
1
1
1
1
1
1
0
0
0
I
Fig. 4. A typical individual in the GA population.
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ALGORITHM II
IMPLEMENTATION OF GA FOR BESS SCHEDULING
Step 1: The initial population is randomly generated by
considering only −1, 0, and 1. To improve the algorithm’s
efficiency, an additional individual is created and inserted
into the population (B in Fig. 4 and (16)). This individual is
designed by taking into account the hourly profile of the
FEP; if the price at determined hour is lower than the daily
average, the charging operation is suggested by assigning 1
to the element of this individual ( =1); otherwise, if the
price is higher than the daily average, −1 is assigned
( =−1); finally, if the price is equal to the daily average, no
control action is considered, by assigning 0 to the
corresponding element ( =0). Besides of those individuals
randomly generated this additional individual is added
improving the general performance of the algorithm.
Step 2: The DNC is evaluated for each individual in the
population using (18), which depends on the FEP and the
scheduled power of the LAB at each time step. Once the
DNC of each individual has been calculated, all of them are
sorted in a list according to their DNCs ( , ⦡ = 1, … , ),
from the smallest to the largest value; finally, according to
this order, the fitness of each individual is assigned by
using (19):
In the PRC strategy, it is assumed that the reduction of the
BESS-aggregator power is carried out according to the
capacity of each BESS; in other words, if a reduction in the
power transaction between the BESS aggregator and SG is
required, those BESSs with higher capacities reduce their
power to be charged or discharged at higher magnitudes than
the other BESSs of lower capacities.
This idea is mathematically expressed in (22), where the
weighting factor ( ⦡ = 1, … , ) reflects the reduction
degree of each BESS. The power reduction of each BESS is
effectively applied by setting the maximum current ( ) to a
convenient value.
⦡
=
With respect to the UPR strategy, it is assumed that the
reduction of the BESS-aggregator power is uniformly carried
out by all BESSs integrated in the aggregator; in this way, the
power reduction and consequently its effects are equally
distributed over the enrolled BESSs.
This idea is implemented by assigning the value of each
weighting factor ( ⦡ = 1, … , ) according to (23):
= 1⁄ ⦡
( + 1) −
=
⦡ = 1,2, … , . (19)
∑ ( +1 − )
Step 3: Reproduction, crossing, and mutation operators are
applied; hence, the population of the next generation is
defined. Frequently, high values of the crossover rate are
used ( → 1), while very low values of the mutation rate are
typically used ( → 0).
Regarding the BESS aggregator, Fig. 5 presents its general
structure. As can be observed, the aggregator is composed of
several BESSs similar to that previously shown in Fig. 1,
which operates in a decentralized way; the BMS of each BESS
optimizes its own operation using an FEP provided by the
aggregator, which is responsible for guaranteeing that the total
power from or to the aggregated BESS ( , ) is within the
corresponding limit defined by the Distribution Network
Operator (DNO); it is then designated as ( ).
In this way, the optimal scheduling of each BESS is added
as presented in (20); then, the excess power required or
supplied by the BESS aggregator (
) is estimated according
to (21), where the limit imposed by the SG is included.
=
=
,
,
⦡ = 1, … , ;
⦡ = 1, … ,
−
= 1,2, … , .
(23)
Once the actual condition of the BESS aggregator has been
defined through variable (
), excess operating power is
eliminated by applying Algorithm III combined with (22) or
(23) depending on the strategy to be used (PRC or UPR,
respectively). Both approaches take advantage of the
communication infrastructure of the SG to accommodate the
BESS aggregator power with regard to the technical
capabilities of the distribution system.
As in Algorithm III (specifically in (24) and (26)), the
maximum charging power is required. It is calculated by using
the regulation voltage (with a typical value equal to 2.23
V/cell) because this is the maximum operational voltage at
which the charge controller would operate the LAB bank; that
is the reason =2.23 V/cell is included in these equations.
Maximum Charging Current
BMS
Power
System
,
= 1,2, … , . (22)
BESS
Aggregator
Control
Signal n
LAB
Storage
System 1
Maximum Charging Current
BMS
LAB
Storage
System n
(20)
;
(21)
To remove this excess in the power transaction, two
mechanisms have been evaluated: Power Reduction by
Capacity (PRC) and Uniform Power Reduction (UPR).
Maximum Charging Current
BMS
LAB
Storage
System N
Fig. 5. General architecture of a BESS aggregator.
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Step 1: If {
> 0}, go to step 2, or, otherwise, stop.
Step 2: Calculate the maximum charging power allowed for
each BESS; this is estimated by using (24):
=(
)
⦡
= 1,2, … , . (24)
Step 3: Estimate the reduction in the maximum charging or
discharging power allowed for each BESS by means of (25):
= (1 −
)
⦡
= 1,2, … , . (25)
Step 4: Calculate the new limit of the maximum charging or
discharging current of each BESS using (26):
=
⁄
⦡
= 1,2, … , . (26)
Step 5: Using the value of
previously assigned in Step 4,
evaluate the performance of each BESS and the power
transaction between the BESS aggregator and SG ( , ⦡ =
1, … , ); estimate
according to (21); and go to Step 1.
IV. CASE S TUDY
The comprehensive methodology developed in this work for
the optimal management of the BESS is illustrated in this
section by analyzing the independent and aggregated operation
using data of the Spanish electricity market of the years 20132016 [35]. This is done to evaluate the influence of forecasting
error and power-system limitations over a representative time
period. Sub-section IV-A describes the results obtained from
the FEP process, and IV-B and IV-C evaluate the capabilities
of the proposed method for controlling an independent BESS
and a BESS aggregator. Finally, in IV-D, the impact of
forecasting error and the SG capacity are statistically
analyzed. All numerical experiments have been implemented
in MATLAB® in a computer with i7-3630QM CPU at 2.40
GHz, 8GB of RAM, and a 64-bit operating system.
A. Electricity Price Forecasting using Spanish Market Data
The FEP was performed by applying the process described
in III-A based on time-series analysis. Two HWEP databases
were used; the first database corresponds to the time period of
January 1 to February 7, 2016, measured on an hourly basis.
This time series was used to carry out the corresponding FEP
for February 8, 2016, using the discrete functions of
and
with 150 intervals. Thus, the resulting information served as
an input to the optimization analysis under independent and
aggregated modes for this day. The second database
corresponds to the time period of December 1, 2013, to
December 31, 2015; this database was used to analyze the
influence of forecasting error and the SG capacity on the
performance of BESS operation from a statistical perspective.
Regarding the ARMA model, a maximum of 30
coefficients was considered for autoregressive and movingaverage parameters, selecting the maximum number of
parameters between those suggested by the BIC and AIC
criteria. As described earlier, the first database was used to
perform time-series analysis for February 8, 2016; thus, the
ARMA coefficients selected ( =5 and =2) are shown in
Tables I and II. With respect to statistical checking,
(α=0.05)=106.394 and =34.036 with 84 degrees of
freedom ( − − =84) were calculated. Fig. 6 shows the
obtained FEP presenting the actual and forecasted profiles.
Respecting the analysis of the second HWEP database, a
time series of December 2013 was used for the prediction of
January 2014; more precisely, the data of December 2013
were analyzed to estimate the order of the corresponding
ARMA model; then, price forecasting over each day of
January was carried out, updating the values of the ARMA
model coefficients. This process was sequentially repeated
over the years of 2014 and 2015, so the error related to the
FEP was statistically investigated.
The results related to the ARMA model’s coefficients of
the second HWEP database are shown in Table III, while a
histogram of forecasting error is presented in Fig. 7, where the
mean and standard deviation are estimated as 0.9711 €/MWh
and 9.5224 €/MWh, respectively. The incorporation of these
data in optimal BESS operation is carefully described in
subsections IV-B, IV-C, and IV-D.
B. Operation of an Independent BESS
This sub-section illustrates the operation of an independent
BESS with the structure shown in Fig. 1 ( =1) conformed by
2V-cells of
=1000 Ah with
=0.3, an initial SOC
equal to 0.3 (
=0.3), 200 serial-connected cells ( =200),
and one string ( =1). The maximum charging or discharging
current per cell was adjusted to 100 A/cell ( = ). The
actual electricity price and the FEP of February 8, 2016, as
previously described, were jointly used with an integer-coded
GA with 200 individuals ( =200), 100 generations ( =100), a
crossover rate equal to 90% ( =90%), and a mutation rate of
1% ( =1%). The ambient temperature was assumed to be
constant, equal to 298K ( , =298K ⦡ = 1, … ,24), which
means its effects on the LAB are not considered.
TABLE I
AUTOREGRESSIVE P ARAMETERS
2.7199
-2.7750
1.1257
-0.0962
0.0155
TABLE II
MOVING AVERAGE P ARAMETERS
-1.6858
Electricity price (€/MWh)
ALGORITHM III
REDUCTION ON POWER S CHEDULIN G OF BESS AGGREGATOR
50
45
40
35
30
25
20
15
10
5
0
0:00
0.9397
Actual
Forecasted
5:45
11:30
Time (h)
17:15
23:00
Fig. 6. Actual and predicted electricity prices (2/8/2016).
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2000
Objective function (€)
-4.2
-4.4
-4.6
-4.8
-5
-5.301830
-5.2
-5.4
1 10 19 28 37 46 55 64 73 82 91 100
Generations
50
40
30
20
10
0
-10
-20
-30
-40
-50
0:00
BESS Power
5:45
11:30
Time (h)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
SOC
0
17:15
23:00
State of charge
Fig. 8. Convergence of GA (2/8/2016).
Fig. 9. Optimal scheduling of BESS (2/8/2016).
450
1000
440
500
0
-100
-50
0
50
Forecasting error (€/MWh)
100
Fig. 7. Histogram of electricity-price forecasting error.
Fig. 8 shows the convergence of the GA completed in
49.127 seconds. Otherwise, the behavior of the LAB during
charging conditions was estimated by the BM considering
=10-10.
As can be observed, the objective function (DNC) takes
negative values due to the fact that BESS discharging (selling
energy to SG) has been modeled with a negative sign (sign of
the LAB discharging current and power); during the hourly
operation, the amount of energy expected to be sold is higher
than the amount expected to be bought from the SG.
The optimal LAB operation in terms of power transaction,
battery voltage, and SOC is shown in Figs. 9 and 10, where,
initially, BESS is charged during low-price periods in the
early morning, is modestly discharged during late-morning
hours, has a resting period during midday hours, is modestly
charged during the afternoon, has a another resting period
after this, and is heavily discharged at high-price periods
during the night hours, following the trend of the FEP.
BESS Voltage (V)
Frequency
1500
-4
Power (kW)
T ABLE III
ARMA MODELS FOR P RICE FORECASTING D URING 2014 AND 2015.
Year
Month
− −
(α=0.05)
2013
Dec
10
9
204
218.592
238.322
Jan
15
3
205
220.3015
239.4034
Feb
5
9
188
189.1803
220.9908
Mar
14
4
205
200.9628
239.4034
Apr
18
20
178
201.69
210.1298
May
18
14
191
216.7428
224.2446
June
9
20
187
177.649
219.9058
2014
July
14
12
197
230.7463
243.5138
Aug
9
15
199
216.2852
232.9118
Sept
7
8
201
256.5014
235.0765
Oct
10
6
207
248.3143
241.5657
Nov
3
6
207
198.8872
241.5657
Dec
3
16
204
206.8395
238.322
Jan
12
2
209
248.929
243.7272
Feb
6
2
194
252.4932
227.4964
Mar
18
4
201
202.6716
235.0765
Apr
6
4
206
178.048
240.4847
May
17
13
193
198.7169
226.4127
2015
June
7
6
203
233.7341
237.2404
July
6
16
201
251.1403
235.0765
Aug
17
9
197
188.617
230.7463
Sept
17
4
195
130.6394
228.5799
Oct
5
4
214
249.264
249.1275
Nov
10
20
186
170.1237
218.8205
430
420
410
400
390
0:00
5:45
11:30
Time (h)
17:15
23:00
Fig. 10. Voltage of BESS (2/8/2016).
As the limit of the distribution system is not being taken
into account in this section (unconstrained operation), a nonstrategy for BESS power reduction is applied. The forecasted
DNC is estimated as -5.301830 € (obtained from the
implementation of the proposed GA shown in Fig. 8), while
the actual DNC is -4.192483 € (obtained from the application
of the management strategy suggested by the GA over the
actual price signal, i.e., evaluating (18) using the actual prices
instead of the forecasted ones). This result shows a reduction
in the expected revenue from the power transaction between
the LAB and SG of €1.109, which is around 21% and is
directly related to the forecasting error of electricity prices.
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C. Operation of a BESS Aggregator
2000
In this sub-section, a BESS aggregator with 15 systems, and
with the architecture presented in Fig. 5 ( =15), is analyzed
by considering the price forecasting calculated in IV-A (the
FEP of Fig. 6) and the optimization parameters used in IV-B.
The data for each system are presented in Table IV with
=0.3 ⦡ = 1, … ,15, while the maximum capacity of
the SG imposed by the DNO is considered to be 1000 kW
( =1000).
1500
Using this information, the strategies of PRC and UPR are
analyzed and discussed.
Regarding the PRC strategy, from the application of the
proposed algorithm described in III-C (Algorithm III with
⦡ = 1, … , defined according to (22)), the scheduling
shown in Fig. 11 was reached; it is possible to observe how
the proposed methodology produces a feasible schedule from
the perspective of the DNO. To obtain these results, an
iterative process of reduction in the maximum battery current
( ⦡ = 1, … ,15) was carried out as presented in Fig. 12 and
Table VI (second column). The entire procedure took only
three iterations; on the one hand, the proposed methodology is
computationally efficient due to the fact that those BESSs with
high contributions to the SG overload experience a higher
reduction in their maximum power transaction, which speeds
up the convergence of the approach; on the other hand, it is
computationally fast because the GA is computed on the BMS
of each BESS in a decentralized manner.
Initially, a 10h-current per cell was assumed to be a
maximum limit ( = = /10); the overload of the SG is
shown in Table VI in the first row and second column; to
remove this unacceptable condition,
was sequentially
reduced until the values shown in Table VIII (the second
column) and the white bars of Fig. 12 where reached.
Table V presents the estimation of the revenue and a
comparison between the constrained and unconstrained
conditions, observing a considerable reduction in the
economic benefit due to SG limitations and forecasting error.
As the calculation process is carried out in a decentralized
way, the computational time per iteration is similar to that
obtained during the independent BESS operation (reported in
IV-B).
Power (kW)
1000
500
0
Charging
limit
Constrained
-500
-1000
-1500
-2000
0:00
Discharging limit
5:45
11:30
17:15
Time (h)
23:00
Fig. 11. Optimal operation of BESS aggregator (2/8/2016) (PRC).
150
Maximum current (A/cell)
T ABLE IV
CHARACTERISTICS OF BESSS INTEGRATED TO THE AGGREGATOR
BESS (kWh)
1
1000
150
15
502
4500
2
300
50
10
34
300
3
500
15
5
9
75
4
250
75
15
63
562.5
5
1500
100
20
669
6000
6
550
30
25
92
825
7
230
60
50
154
1380
8
110
55
18
25
217.8
9
750
15
5
13
112.5
10
300
10
16
11
96
11
270
100
14
85
756
12
450
5
10
6
45
13
140
15
40
19
168
14
550
35
35
151
1347.5
15
850
10
35
67
595
Non-constrained
Iteration 1
Iteration 2
Iteration 3
100
50
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
BESS system
Fig. 12. Maximum current per cell for each BESS (2/8/2016) (PRC).
TABLE V
ECONOMIC PERFORMANCE OF BESS AGGREGATOR (PRC S TRATEGY)
Daily Net Cost (€)
Solution
Forecasted
Actual
Constrained
-138.124144
-109.811260
Unconstrained
-217.344787
-165.666463
With respect to the UPR strategy, from the application of
the proposed algorithm described in III-C (Algorithm III with
⦡ = 1, … , defined according to (23)), the scheduling
shown in Fig. 13 was reached; it is possible to observe how
the proposed methodology produces a feasible schedule from
the perspective of the DNO. To obtain these results, an
iterative process of reduction in the maximum battery current
( ⦡ = 1, … ,15) was carried out as presented in Fig. 14 and
Table VIII. The entire procedure took only eight iterations.
As previously explained, a 10h-current per cell was
assumed to be a maximum limit; the overload of the SG is
shown in Table VI in the first row and third column. To
remove this unacceptable condition,
was sequentially
reduced until the values shown in Table VIII (the third
column) and the gray bars of Fig. 14 where reached.
Table VII presents the estimation of the revenue and a
comparison between the constrained and unconstrained
conditions, observing a considerable reduction in the
economic benefit due to SG limitations and forecasting error.
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The PRC and UPR strategies could be compared by
analyzing the obtained results; from the comparison between
the results presented in tables V and VII, an improvement of
8.3% of the UPR over the PRC using the FEP and 7.4% using
actual prices could be observed. This fact can be observed
qualitatively in figs. 11 and 13, where the amount of power
charged and discharged to the SG have important differences,
being higher when the UPR strategy is applied.
However, another relevant result arises when Figs. 12 and
14 are compared, and Table VI is analyzed because the PRC
strategy has higher computational efficiency, as it is able to
remove the overload of the SG more quickly (in fewer
iterations) than the UPR strategy. This characteristic could be
particularly useful for managing charging or discharging
power during emergency conditions or when the operating
conditions of the SG are under risk.
2000
Non-constrained
1500
Power (kW)
1000
500
Charging
limit
Constrained
0
-500
-1000
-1500
Discharging limit
-2000
0:00
5:45
11:30
17:15
Time (h)
23:00
Fig. 13. Optimal operation of BESS aggregator (2/8/2016) (UPR).
Maximum current (A/cell)
150
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Iteration 6
Iteration 7
Iteration 8
100
50
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
BESS system
Fig. 14. Maximum current per cell for each BESS (2/8/2016) (UPR).
T ABLE VI
CONVERGENCE OF THE P ROPOSED METHODOLOGY
(PRC)
(UPR)
1
700.981549652121
700.9815
2
322.248988901942
579.4792
3
58.0583356121276
488.557
4
-------403.5581
5
-------290.3378
6
-------208.3285
7
-------122.3642
8
-------50.25158
T ABLE VII
ECONOMIC PERFORMANCE OF BESS AGGREGATOR (UPR S TRATEGY)
Daily Net Cost (€)
Solution
Forecasted
Actual
Constrained
-156.208573
-122.114593
Unconstrained
-217.344787
-165.666463
T ABLE VIII
MAXIMUM C URRENT PER CELL FOR E ACH BESS AND STRATEGY
(PRC)
(UPR)
1
39.704433
57.58299
2
28.437849
17.274897
3
49.340390
28.791495
4
22.596897
14.395748
5
40.559986
86.374485
6
47.366544
31.670645
7
17.835719
13.244088
8
10.582127
6.334129
9
73.519157
43.187243
10
29.494046
17.274897
11
23.551881
15.547407
12
44.643180
25.912346
13
13.588557
8.061619
14
42.917748
31.670645
15
76.374089
48.945542
D. Joint Effects of Forecasting Error and System Capacity
In this sub-section, the effects of price forecasting error and
SG limitations, in terms of the maximum power to be sold or
bought by an independent or aggregated BESS, are
statistically analyzed.
Our analysis is carried out by evaluating the performance of
the independent BESS previously described in IV-B (2V-cells
of
=1000 Ah,
=0.3, =200, =1) over the years
of 2014 and 2015, considering different values of the
maximum charging or discharging current, specifically
={0.2( ), 0.4( ), 0.6( ), 0.8( ), }. The FEP and
actual values used in this analysis were those previously
estimated and described in IV-A (specifically, those estimated
by using the second HWEP database of the period of
December 2013 to December 2015).
As stated before in IV-C, a determined BESS ( ) integrated
with a determined aggregator reduces its charging or
discharging power to guarantee the successful operation of the
BESS aggregator from the point of view of the DNO;
depending on the operating conditions and the SG
characteristics, this reduction could be considerable or not. To
evaluate the perspective of each BESS incorporated in the
aggregator (with high and low reduction of the charging or
discharging power), an independent BESS constrained to
different values of
(specifically between 20% and 100% of
) has been investigated.
The most important results related to this experiment are the
predicted economic benefit (obtained by evaluating the
scheduling obtained from the GA using the FEP), the actual
estimated benefit (obtained by evaluating the near-optimal
solution suggested by the GA on the actual electricity prices),
and the corresponding estimation error (estimated as the
subtraction between the forecasted and actual benefit).
The corresponding results obtained from the estimation of
all of these variables are presented as histograms of frequency
in Figs. 15-17, while the mean and standard deviation values
are shown in Tables IX, X, and XI. In Fig. 15, all forecasted
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Transactions on Smart Grid
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Frequency
400
20%(Imax)
40%(Imax)
60%(Imax)
80%(Imax)
100%(Imax)
200
100
-10
-5
Predicted benefit (€)
0
Fig. 15. Histogram of forecasted daily net cost.
700
600
Frequency
500
400
20%(Imax)
40%(Imax)
60%(Imax)
80%(Imax)
100%(Imax)
300
200
100
0
-20
-15
-10
-5
Actual benefit (€)
Fig. 16. Histogram of actual daily net cost.
400
20%(Imax)
40%(Imax)
60%(Imax)
80%(Imax)
100%(Imax)
300
200
100
0
-15
-10
-5
0
5
10
Difference between actual
Costand
(€)predicted benefits (€)
Fig. 17. Histogram of estimation error on daily net cost.
Mean
Dev.
T ABLE IX
MEAN AND S TANDARD DEVIATION OF FORECASTED DNC
0.2( )
0.4( )
0.6( )
0.8( )
-1.041
-1.96896
-2.75785
-3.35447
-3.6694
0.553846
1.118292
1.683701
2.192115
2.511537
Mean
Dev.
TABLE X
MEAN AND STANDARD DEVIATION OF ACTUAL DNC
0.2( )
0.4( )
0.6( )
0.8( )
-0.8469
-1.5945
-2.1973
-2.62869
-2.8732
0.559271
1.123313
1.682101
2.133537
2.441735
V. CONCLUSIONS
300
0
-15
500
T ABLE XI
MEAN AND S TANDARD DEVIATION OF ESTIMATION ERROR ON DNC
0.2( )
0.4( )
0.6( )
0.8( )
Mean
-0.1941
-0.37446
-0.56055
-0.72578
-0.7962
Dev.
0.519521
1.011225
1.497099
1.887078
2.084716
600
500
600
Frequency
DNCs are negative to maximize the economic benefit.
However, according to the results shown in Fig. 16, when
actual values are evaluated, some positive values arise as a
consequence of the forecasting error, which represents a total
loss of revenue due to the difference between the FEP and the
actual values. Similarly, some values of the DNC highly
influenced by forecasting error appear as negative estimation
errors in Fig. 17 due to the fact that in some days, the revenue
estimated using the FEP was higher than that obtained when
the actual values were finally evaluated.
In addition, the reduction of the economic benefit could be
observed in the first rows of Tables IX and X, in which the
mean values of the forecasted DNC are higher than those of
the actual DNC due to the fact that we always expect a higher
economic benefit than that which occurs when the actual
values arise. This fact directly influences the planning problem
and the expected revenue during the lifetime of the system and
should be considered during the planning evaluation process.
Regarding the SG limits, it is possible to observe how, as
the maximum battery current is reduced (to accommodate the
BESS power into the SG), the economic benefit and its
variability are considerably reduced. In a general sense, those
BESSs with low reductions of their charging or discharging
capabilities are going to observe reductions of their economic
benefits mainly influenced by forecasting error. On the
contrary, those BESSs with high reductions of their
capabilities are going to observe reductions of their economic
benefits mainly influenced by the SG power limits.
0
5
In this paper, a comprehensive methodology for the optimal
management of the LAB, operating both independently and as
an aggregated system in the RT pricing environments, has
been developed and illustrated through some representative
examples. The proposed methodology is based on solving the
optimization problem using an integer-coded GA that is
formulated to minimize the DNC, thereby allowing the
incorporation of the non-linear characteristics of the LAB,
which are related to voltage-current behavior and gassing
phenomena, charge regulation, and the non-linear efficiency of
a bidirectional conversion system. Moreover, the technical
limitations of the SG imposed by the DNO, which are related
to the maximum system capacity, have been taken into
account. Hence, the proposed methodology works in a
decentralized manner, demonstrating excellent computational
efficiency and the ability to provide a feasible and reliable
solution from the perspective of the DNO, thus guaranteeing
the safe incorporation of the BESS on a massive scale, which
is of major importance. Using a representative amount of data
from the Spanish electricity market, the effects of forecasting
error and the SG limitations on the performance of a BESS
aggregator have been evaluated, concluding that the economic
benefit of those BESSs with relatively low reductions of their
power-transaction capabilities is mainly influenced by
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Transactions on Smart Grid
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forecasting error, while the economic benefit of those BESSs
with relatively high reductions of their power-transaction
capabilities is mainly influenced by the loadability of the SG.
In a general sense, the obtained results reveal the importance
of considering the specific type of BESS to be installed (in our
case, the LAB), all of its associated components (charge
controller and power converter) and the optimization
technique used to analyze its behavior, the specific
competition rules among the BESSs integrated in the
aggregator (power reduction by capacity or uniform power
reduction), and the mathematical tool used to carry out the
FEP (the ARMA model) and its associated error, in the
optimal operation of BESSs from an economic perspective.
Investments in the expansion and planning of distribution
systems directly improve the economic performance of the
BESSs integrated with it. Hence, governmental incentives to
promote the massive deployment of BESSs should be
accompanied by important investments in distribution-system
expansion.
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1949-3053 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Smart Grid
15
Juan M. Lujano-Rojas received his Ph.D. degree from
the University of Zaragoza, Spain, in 2012. In 2013, he
joined the University of Beira Interior (UBI), Covilha,
Portugal, as a researcher in the EU-funded FP7 project
SiNGULAR. Currently, he is a post-doctoral fellow at
INESC-ID/IST-UL and UBI.
Rodolfo Dufo-López received the B.S., M.S. and Ph.D.
degree from University of Zaragoza, Spain, in 1994, 2001
and 2007, respectively. In 2004, he joined the University
of Zaragoza, where he is currently a Professor of the
Department of Electrical Engineering. He has published
more than 50 journal papers and conference proceedings.
His research interests include renewable energy,
distribution systems, smart grid, power quality,
generation and demand side management, and power systems analysis.
José L. Bernal-Agustín (M’04) received his Ph.D.
degree from the University of Zaragoza, Spain, in 1998.
In 1991, he joined the University of Zaragoza, where he
is currently a Professor of the Department of Electrical
Engineering. He has published more than 70 journal
papers and conference proceedings. His research interests
include applications of evolutionary algorithms in
electrical power systems, distribution systems, electric
markets, renewable energy and smart grids.
João P. S. Catalão (M’04-SM’12) received the M.Sc.
degree from the Instituto Superior Técnico (IST),
Lisbon, Portugal, in 2003, and the Ph.D. degree and
Habilitation for Full Professor ("Agregação") from the
University of Beira Interior (UBI), Covilha, Portugal, in
2007 and 2013, respectively.
Currently, he is a Professor at the Faculty of
Engineering of the University of Porto (FEUP), Porto,
Portugal, and Researcher at INESC TEC, INESC-ID/IST-UL, and
C-MAST/UBI. He was the Primary Coordinator of the EU-funded FP7 project
SiNGULAR ("Smart and Sustainable Insular Electricity Grids Under LargeScale Renewable Integration"), a 5.2-million-euro project involving 11
industry partners. He has authored or coauthored more than 470 publications,
including 155 journal papers, 281 conference proceedings papers, 23 book
chapters, and 11 technical reports, with an h-index of 27 and over 3000
citations (according to Google Scholar), having supervised more than 45 postdocs, Ph.D. and M.Sc. students. He is the Editor of the books entitled Electric
Power Systems: Advanced Forecasting Techniques and Optimal Generation
Scheduling and Smart and Sustainable Power Systems: Operations, Planning
and Economics of Insular Electricity Grids (Boca Raton, FL, USA: CRC
Press, 2012 and 2015, respectively). His research interests include power
system operations and planning, hydro and thermal scheduling, wind and price
forecasting, distributed renewable generation, demand response and smart
grids.
Prof. Catalão is an Editor of the IEEE TRANSACTIONS ON SMART GRID , an
Editor of the IEEE TRANSACTIONS ON S USTAINABLE ENERGY , and an
Associate Editor of the IET Renewable Power Generation. He was the Guest
Editor-in-Chief for the Special Section on "Real-Time Demand Response" of
the IEEE TRANSACTIONS ON SMART GRID, published in December 2012, and
the Guest Editor-in-Chief for the Special Section on "Reserve and Flexibility
for Handling Variability and Uncertainty of Renewable Generation" of the
IEEE TRANSACTIONS ON S USTAINABLE ENERGY , published in April 2016. He
was the recipient of the 2011 Scientific Merit Award UBI-FE/Santander
Universities and the 2012 Scientific Award UTL/Santander Totta. Also, he
has won 4 Best Paper Awards at IEEE Conferences.
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